Chemistry Reference
In-Depth Information
only configurations of given S and M
S
belonging to a given molecular
symmetry, have nonzero matrix elements of the molecular Hamiltonian.
Even if the method is, in principle, exact if we include all configurations,
expansion (8.1) converges usually quite slowly and the number of
configurations becomes rapidly very large, involving up to millions of
determinants.
1
This is due to the difficulty of the wavefunction (8.1) in accounting for
the cusp condition for each electron pair (Kutzelnigg, 1985):
¼
1
Y
0
@
Y
1
2
lim
r
ij
!
0
ð
8
:
2
Þ
@
r
ij
which is needed to keep the wavefunction finite when r
ij
¼
0 in presence
of the singularities of the Coulomb terms in the Hamiltonian.
Following Kato (1957), Kutzelnigg (1985) has shown that the
Y
expanded in powers of the interelectronic distance r
ij
ð
1
þ
ar
ij
þ
br
ij
þ
cr
ij
þ Þ
Y
¼
Y
ð
8
:
3
Þ
0
satisfies this condition near the singular points for the pair of electrons
i and j when a
¼
1
2
. In fact:
@
Y
@
r
ij
¼
Y
ð
a
þ
2br
ij
þ Þ
ð
8
:
4
Þ
0
1
Y
0
@
Y
@
r
ij
¼
a
þ
2br
ij
þ
ð
8
:
5
Þ
¼
a
¼
1
Y
0
@
Y
1
2
lim
r
ij
!
0
ð
8
:
6
Þ
@
r
ij
where a is a constant, with a
¼
2
if i and j are both electrons, a
¼
Z
B
if i
is a nucleus of charge
þ
Z
B
and j an electron.
Using He as a simple example, Kutzelnigg (1985) showed that use of
a starting wavefunction of the type
1
1
2
r
12
0
1
þ
ð
8
:
7
Þ
Y
1
Special techniques are required for solving the related secular equations of such huge dimen-
sions (Roos, 1972).
